Four Practical Men

In 1907 Herbert Chatley, lecturer in applied mechanics at Portsmouth Tech­nical Institute, published The Problem of Flight: A Text-Book of Aerial Engi­neering. The book went through two further editions, in 1910 and 1921. In the preface Chatley explained that, in terms of mathematics, he followed the practice, “well established in engineering,” of omitting factors that appear unimportant: “The formulae are therefore ‘engineering formulae’ in the strict sense of the word, i. e. they are not the result of a deep mathematical analysis which it is, in the majority of cases, almost impossible to apply.”67

Chatley’s account of lift was eclectic. It involved elements of perfect fluid theory and discontinuity theory along with the Newtonian analysis and its problematic sin2 formula. To overcome the difficulties he added various em­pirical corrections. These cut across the deductive links between the formulas in a manner that might have been calculated to offend the sensibilities of a wrangler. He began by mentioning the continuous flow of a perfect fluid over an inclined plane. This, said Chatley, is described in Lamb’s Hydrody­namics, and its reality has been demonstrated by photographs taken by Prof. Hele-Shaw. (In fact Hele-Shaw’s photographs show the behavior of a viscous fluid in slow motion between two glass plates that are very close together. Stokes was able to show that, under these conditions, “creeping motion,” as it is called, provides an accurate simulation of the flow of a perfect fluid. The forces at work and the boundary conditions are different, but the pho­tographs show what a perfect fluid flow would look like.)68 Chatley went on to assert that at greater speeds this flow breaks down and is replaced by one showing surfaces of discontinuity enclosing pockets of turbulence on the rear of the plate. These eddies reduce the pressure. So far, Chatley’s qualitative pic­ture is approximately that of Kirchhoff-Rayleigh flow, combined with some of Kelvin’s ideas about the turbulence in the dead-water region.

Chatley then introduced some mathematics and deduced Newton’s sin2 formula in the manner just described. What Chatley meant by an engineer­ing formula became clear when he asserted (without giving evidence) that the effect of the eddying on the rear of the plate is to augment the pressure by half as much again. He therefore repeated the above formula, but now multiplied by 3/2, and called it Nmax.. Chatley claimed that this formula agrees “very fairly” with some experimental results of Coulomb, although it was conceded that, for small angles, experimenters disagree greatly. In what is presumably a reference to Rayleigh’s paper, he went on: “The latest results are almost unanimous in making the variations of thrust as sin0 and not as sin2 0. All these following expressions are thus divided by sin 0” (29). Thus the sin2 0 term in the Newtonian formula was simply altered to sin0. In the 1921 edition this abrupt step was justified by saying that the original formula is approximately correct for large angles of incidence (from 60° to 90°), but for small angles, “owing to the continuity of the air, sin0 must be substituted for sin2 0” (31). In the 1910 and 1921 editions the reader was told that the end result is “practically correct” for plane surfaces and, with “slight correction to the coefficients,” also applies to curved surfaces.

Algernon Berriman was the chief engineer at the Daimler works in Cov­entry and the technical editor of Flight. In 1911 both Flight and Aeronautics published accounts of his lectures titled “The Mathematics of the Cambered Plane,” and in 1913 he published Aviation: An Introduction to the Elements of Flight.69 This book was based on lectures he had given at the Northampton Polytechnic Institute. Berriman also started from the Newtonian idea of ac­tion and reaction. Lift came from the reaction on the wing of the mass of air that was, by some means, forced downward by the wing. Thus, “the wing in flight continually accelerates a mass of air downwards, and must derive a lift therefrom.”70 The basic formula is Force = Mass X Acceleration, but how is this formula to be applied? What is the mass of air that is involved? The original Newtonian picture must have underestimated this mass, hence the underestimation of the lift that can be generated.

Berriman assumed that the wing sweeps out, and pushes down, a greater area than is suggested by the simple geometry of an inclined plane. The wing exerts an influence on “all molecules within an indefinite proximity to the plane; in other words a stratum of air of indefinite depth.”71 Instead of stating that a wing of area A engages with a quantity of air determined by the frontal projection A sin0, the assumption was made that it sweeps out a larger area, AS, whose sweep factor S is typically much larger than sin 0. Berriman said that “practical considerations” gave reason to believe that “the effective sweep of a cambered plane may be defined in terms of the chord of the plane” (5). In other words, Berriman put S = 1. This equation was based on the experience of the pioneers and experimenters, such as Langley, who found by trial and error that they got the best results with a biplane when they positioned one wing about one chord length above the other.72 The reasoning was hardly rig­orous, but the assumption allowed Berriman to avoid the troublesome sine – squared term.

Although the concept of “sweep” was popular with the practical men, it had few practical advantages. All that could be done was to determine the lift empirically and then deduce that the quantity called sweep must have such and such a numerical value. No one could go in the other direction. There was no way to predict the lift from the sweep. One might argue, inductively, that similar wings will have similar sweeps, but one could also say that similar wings have similar lifts, so in practice nothing is gained by introducing the concept. G. H. Bryan, after expressing irritation with Berriman’s casual way with trigonometric formulas,73 identified the source of the difficulty:

there is no such thing as “sweep” except in Newton’s ideal medium of non­interfering particles satisfying the sine squared law. In a fluid medium the disturbance produced by a moving solid theoretically extends to an infinite distance, gradually decreasing as we go further off. Mr. Berriman’s “sweep” is, physically speaking, an impossibility. If, however, “sweep” is defined as the depth of a hypothetical column of air, the change of momentum in which would represent the pressure on the plane, then the introduction of this new quantity is only a useless and unnecessary complication. Instead of facilitating the determination of the unknown data of the problem, it merely replaces one variable which is physically intelligible and capable of experimental determi­nation by another variable satisfying neither of these conditions. (265)

The practical men never found a way to use the idea of sweep so that they could sustain what Kuhn called a “puzzle-solving tradition.”74

The work reported in Albert Thurston’s Elementary Aeronautics of 1911 was based on the hope of identifying the significant properties of the flow, such as the sweep of a wing, in an empirical manner.75 Thurston had worked for Sir Hiram Maxim and then became a lecturer in aeronautics at the East London Technical College. He took numerous photographs of airflows made visible by jets of smoke as they streamed past objects of various shapes. The objects ranged from rectangular blocks to aerofoil shapes, or “aero-curves” as Thurston called them. He concluded that the important qualitative factor in the flow over a wing was that the entry at the leading edge was smooth and avoided the “shock” detectable in the case of a simple, flat plate. The avoid­ance of shock was possible because of the rounded and slightly dipped front edge characteristic of a wing, a shape whose advantages had been discovered empirically by Horatio Phillips and Otto Lilienthal.76 The essential thing, ac­cording to Thurston, was to maintain a smooth “streamlined” flow. The at­tempt to impose sudden changes in the velocity of the air merely produces surfaces of discontinuity (20). The photographs showed that with a good, winglike shape at small angles of incidence, “the air divides at the front edge and hugs both sides as it passes along; its resistance to change of motion caus­ing a compression on the lower side of the plane and a rarefaction or suction on the upper side. As the inclination is increased a critical angle appears to be reached, after which the stream line ceases to follow the upper side and forms a surface of discontinuity with corresponding eddies” (21-24).

As with the work of Eden, Bairstow, and Melvill Jones at the NPL, such photographs revealed that the model of discontinuous flow was not going to provide a basis for understanding lift. The photographs did, though, support ideas about the extended sweep of a wing. Thurston avoided using the word “sweep” but referred to the “field” of a wing and, on the basis of his photo­graphs, asserted: “The air affected by an aeroplane [= wing], that is the field of an aeroplane, is greater than the air lying in its path. Thus. . . it will be seen that air, which is considerably above the front edge of the plane, is within the range of the plane, and is deflected downwards” (26-27).

Even with photographs of smoke traces in the flow, it proved impossible to identify the sweep in a quantitative way, and the underlying causes of the qualitative effects visible in the photographs remained obscure. Thurston, however, was convinced that the secret of good design, both for wings and other components, was attention to streamlining, that is, ensuring that the lines of flow in the immediate neighborhood of the body coincide with the surface of the body. Like Chatley, Thurston drew on the work of Hele-Shaw to show how streamline flow works and what it looks like.77

Frederick Handley Page also appealed to Hele-Shaw’s photographs. Hand­ley Page was one of the celebrated pioneers in the British aviation industry. Despite its early inability to deliver BE2s, his firm was later to achieve fame for its manufacture of large bomber and passenger aircraft. In April 1911 he gave a lecture at the Aeronautical Society titled “The Pressure on Plane and Curved Surfaces Moving through the Air.”78 He began by discussing the re­sult that had caused so much difficulty for the discontinuity theory, namely, that the formation of “dead” air behind the plate happens when it has passed the critical angle. Practical aeronautics by contrast, said Handley Page, deals with small angles of incidence where the flow hugs the back of the plate or aerofoil. There is then a maximum of lift to drift and a minimum of eddy disturbance. Commenting on a Hele-Shaw photograph of the flow of a per­fect fluid past an inclined plane, Handley Page said, “The air on meeting the plane divides into two streams. . . the streams meeting again at the back of the plane. At high velocities the eddies and turbulence at the rear of the plane completely obscure this, but up to the critical angle at which the ‘live’ air stream leaves the plane back, the effect is still the same” (48).

Handley Page, like Chatley and Thurston, took ideal fluid theory to pro­vide an accurate picture of real flow round a wing, even when there are no surfaces of discontinuity or other complications such as vortices in the flow. Hele-Shaw’s photographs, however, depicted d’Alembert’s “paradox” in ac­tion, not a wing delivering lift. The practical men were thus walking into the trap that Greenhill had identified in his lectures at Imperial. They were proposing a picture of the flow which the mathematician would immediately recognize as one that gave neither lift nor drag.

No one in the audience at the Aeronautical Society mentioned this prob­lem in the subsequent discussion. Even if the point had been raised it would have had little impact on Handley Page’s eclectic argument because, immedi­ately after this appeal to hydrodynamic theory, the perspective was changed. He adopted the neo-Newtonian approach but suggested refining the idea of sweep by dividing it into two parts. This modification revealed his real inter­est in the Hele-Shaw pictures. The portrayal of the flow at the leading edge suggested to Handley Page that two different processes were at work. There was the sweep associated with the flow upward from the stagnation point to the leading edge, and the sweep associated with the downward flow toward the rear edge. This complication enabled him to refine the mathematics of the sweep picture, but it did not get round Bryan’s objections: it merely doubled the number of unknowns. Handley Page still had to infer the total sweep from the observed lift and had no way to apportion the contributions of the two components of the sweep that he postulated.

Handley Page’s lecture was generally well received, though no mathemati­cians contributed to the discussion—if, indeed, any were present. Cooper, who had been so scathing about Greenhill, congratulated Handley Page on getting a formula that applied to experimental results: “it is not everybody,” he added, “who does that.” Cooper was either being polite or had failed to see how little had been achieved. He went on to say that he thought Handley Page’s analysis applied more to the flat plate, where the leading edge caused a “shock” in the flow, than it did to an aerofoil with its rounded, dipping front edge. Here, claimed Cooper, the stagnation point would be on the very front, not below the leading edge on the underside of the wing, as it was in Hele – Shaw’s picture of the plate. In what may have been meant, at least in part, as a response to this point, Handley Page said, “It seems to me that the entering front edge is only a kind of transformer. . . a curved plane is more efficient than a flat one because you have a more efficient transformer” (63). Unfortu­nately, no attempt was made to explain the metaphor of the “transformer.”

A few years after this exchange Handley Page introduced the famous, and commercially lucrative, Handley Page wing. The standard wing was modified by introducing a slot along the leading edge which changed the flow at the leading edge by directing air from underneath the front of the wing onto its upper surface. It was, in effect, a small extra aerofoil that ran along the leading edge of the main wing. The resulting change in the flow significantly increased the lift and delayed the stall.79 Could this innovation have been a result of the earlier conception of the leading edge as having a capacity to “transform” the flow? When Handley Page described his invention in the Aeronautical Journal, he made no mention of the metaphor of the transformer. Indeed, he never gave a clear account of the thought processes behind his invention, so the question must remain unanswered. (He had originally tried making slots that ran from the leading edge to the trailing edge, that is, along the chord of the wing rather than along the span. This suggests that the process of inven­tion was trial and error, rather than theory-led.) The value of the leading – edge slot is an example of that intriguing phenomenon “simultaneous dis­covery” and, almost predictably, gave rise to a priority dispute.80 The slot was developed independently in Germany by G. Lachmann, and for many years Thurston also argued his claim to be recognized as the inventor.81